nLab synthetic quasi-coherence

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Contents

Idea

A duality axiom used in various branches of synthetic mathematics, such as

The axiom of synthetic quasi-coherence is a more general version of the Kock-Lawvere axiom in synthetic differential geometry as it can apply to algebras of a generic model of a Horn theory rather than just the Weil algebras of a generic local ring.

Definition

Let 𝕋\mathbb{T} be a Horn theory and let U 𝕋U_\mathbb{T} be the generic model of the theory 𝕋\mathbb{T}. A homomorphism between two models of 𝕋\mathbb{T} is a function which preserves all the signatures of the theory 𝕋\mathbb{T}. An U 𝕋U_\mathbb{T}-algebra consists of a 𝕋\mathbb{T}-model AA with a homomorphism h:U 𝕋Ah:U_\mathbb{T} \to A. An U 𝕋U_\mathbb{T}-algebra homomorphism on an U 𝕋U_\mathbb{T}-algebra AA is a homomorphism from AA to U 𝕋U_\mathbb{T}. The spectrum of AA is the set of U 𝕋U_\mathbb{T}-algebra homomorphisms:

Spec(A)hom U 𝕋Alg(A,U 𝕋)\mathrm{Spec}(A) \coloneqq \mathrm{hom}_{U_\mathbb{T}\mathrm{Alg}}(A, U_\mathbb{T})

An U 𝕋U_\mathbb{T}-algebra AA is quasi-coherent if the canonical evaluation homomorphism

λa.λf.f(a):AU 𝕋 Spec(A)\lambda a.\lambda f.f(a):A \to {U_\mathbb{T}}^{\mathrm{Spec}(A)}

is an isomorphism.

Definition

The axiom SQCP or synthetic quasi-coherence for the univariate polynomial algebra states that the free U 𝕋U_\mathbb{T}-algebra on one generator U 𝕋[x]U_\mathbb{T}[x] is quasi-coherent.

Phoa's principle for distributive lattices is equivalent in strength to the axiom SQCP for U 𝕋U_\mathbb{T}-algebras where 𝕋\mathbb{T} is the Lawvere theory of distributive lattices.

An U 𝕋U_\mathbb{T}-algebra AA is finitely quotiented if there is a natural number nn and two finite families of elements a,b:Fin(n)U 𝕋a, b:\mathrm{Fin}(n) \to U_\mathbb{T} in the generic model such that AA is isomorphic to the quotient algebra U 𝕋/a=bU_\mathbb{T} / a = b.

Definition

The axiom SQCI or synthetic quasi-coherence for finitely quotiented algebras states that every finitely quotiented U 𝕋U_\mathbb{T}-algebra of U 𝕋U_\mathbb{T} is quasi-coherent.

In applications to synthetic algebraic geometry and synthetic (infinity,1)-category theory, one usually considers the notion of synthetic quasi-coherence as applying to finitely presented algebras: A finitely presented U 𝕋U_\mathbb{T}-algebra is an U 𝕋U_\mathbb{T}-algebra AA which is isomorphic to a quotient of the free U 𝕋U_\mathbb{T}-algebra on a finite set of generators.

Definition

The axiom SQCF or synthetic quasi-coherence for finitely presented algebras states that all finitely presented U 𝕋U_\mathbb{T}-algebras AA are quasi-coherent.

In applications to synthetic topology, such as synthetic Stone duality, as well as synthetic domain theory, one usually considers the notion of synthetic quasi-coherence as applying to countably presented algebras: A countably presented U 𝕋U_\mathbb{T}-algebra is an U 𝕋U_\mathbb{T}-algebra AA which is isomorphic to a quotient of the free U 𝕋U_\mathbb{T}-algebra on a countable set of generators.

Definition

The axiom SQCC or synthetic quasi-coherence for countably presented algebras states that all countably presented U 𝕋U_\mathbb{T}-algebras AA are quasi-coherent.

In applications to synthetic differential geometry, one considers Artinian U 𝕋U_\mathbb{T}-algebras, i.e. U 𝕋U_\mathbb{T}-algebras that satisfy the descending chain condition.

Definition

The axiom SQCA or synthetic quasi-coherence for Artinian algebras states that all Artinian U 𝕋U_\mathbb{T}-algebras are quasi-coherent.

The Kock-Lawvere axiom is the axiom of synthetic quasi-coherence for Artinian U 𝕋U_\mathbb{T}-algebras with 𝕋\mathbb{T} the Horn theory of a local ring (hence by the definitions given on this page, every U 𝕋U_\mathbb{T}-algebra is a local ring and every Artinian U 𝕋U_\mathbb{T}-algebra is a Weil algebra).

References

Last revised on June 23, 2025 at 23:41:34. See the history of this page for a list of all contributions to it.